Question: Solve for $x$ : $ 3|x - 6| + 8 = 4|x - 6| + 3 $
Answer: Subtract $ {3|x - 6|} $ from both sides: $ \begin{eqnarray} 3|x - 6| + 8 &=& 4|x - 6| + 3 \\ \\ {- 3|x - 6|} && {- 3|x - 6|} \\ \\ 8 &=& 1|x - 6| + 3 \end{eqnarray} $ Subtract $3$ from both sides: $ \begin{eqnarray} 8 &=& 1|x - 6| + 3 \\ \\ {- 3} && {- 3} \\ \\ 5 &=& 1|x - 6| \end{eqnarray} $ Simplify: $ 5 = |x - 6| $ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ -5 = x - 6 $ or $ 5 = x - 6 $ Solve for the solution where $x - 6$ is negative: $ - 5 = x - 6$ Add ${6}$ to both sides: $ \begin{eqnarray} - 5 &=& x - 6 \\ \\ {+ 6} && {+ 6} \\ \\ -5 + 6 &=& x \end{eqnarray} $ $ 1 = x $ Then calculate the solution where $x - 6$ is positive: $ 5 = x - 6 $ Add ${6}$ to both sides: $ \begin{eqnarray} 5 &=& x - 6 \\ \\ {+ 6} && {+ 6} \\ \\ 5 + 6 &=& x \end{eqnarray} $ $ 11 = x $ Thus, the correct answer is $x = 1 $ or $x = 11 $.